Alternating Series Convergence Test $(-1)^k [e - (1+\frac{1}{k})^k$

Question

Determine if the following alternating series converge.

$\sum_{k=1}^\infty (-1)^k [e - (1+\frac{1}{k})^k$]

I know that $e - (1+\frac{1}{k})^k$ is positive for all x and can see that it is a decreasing function by comparing $e - (1+\frac{1}{k+1})^{k+1}$ and $e - (1+\frac{1}{k})^k$ as well as differentiating the function.

Is this enough for a proving that the series will converge?

Answer 1

It is not enough. Consider $\sum_{k=1}^\infty (-1)^k (1+\frac{1}{k})$, which satisfies the conditions you propose but clearly does not converge.